29 Palms Vehicle Detection

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29 Palms Vehicle Detection

• (what we wanted to do)

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Vehicle Tracking
Vehicles trigger network events
Pattern
ESTIMATES
Events are matched to a dynamic pattern to
determine vehicle parameters
Vehicle speed 17 MPH
Entry time 210904
Regression
Analysis
Event
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One-dimensional problem
p
Each node i has a name ni and a position pi along
the road The nodes dont know their positions
exactly, but we do need some estimate
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Problem restrictions
Assumption 1 There is only one vehicle in the
network at a time This lets us separate events
deal with them independently So an event is
characterized by the time that the vehicle
reaches each node in the network
Assumption 2 The vehicle moves with a constant
velocity v. This means that the relationship
between time and position is linear, p vt.
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Linear formulation

• For each vehicle, there are two parameters
• t0 the time the vehicle passes through point p
  0
• v the speed of the vehicle (positive for
  increasing p)
• Each node contributes one equation, if we combine
  all nodes
• t1 t0 (1/v)p1
• t2 t0 (1/v)p2 etc.
• This can be cast as the equation Ax b with

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Least squares solution
This overdetermined system can be solved using
the least squares solution x (ATA)-1ATb
Matrix inversion is only a 2x2! Akin to fitting
a line through a series of points here our
slope is (1/v) and our t0 is the y-intercept We
can use as many equations (at least two) as we
want each node can poll its neighbours and use
as much information as desired, allowing for
missed readings Messages sent are Hi, I saw
the vehicle at time t and I think that Im at
position p. We dont even care who you are.
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Position update
Once we calculate v, go back and make a new guess
at each pi ti (1/v)pinew t0 pinew
(ti-t0)v Update according to some
non-catastrophic weighted rule like
Better Results As time progresses
Detect Vehicle (fix pis)
Update Positions (fix t0, v)
Make Initial guess For pis
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Extensions?
(Are our restrictions problematic for a more
realistic scenario?) One vehicle at a time
constant velocity requirements can be satisfied
providing we consider a small enough region of
the network at a time The 1-D to 2-D extension
is more difficult matched against a single
pattern, would require more patterns (perhaps one
for vertical and one for horizontal travel at
each node?)